Problem: What number makes this equation true?
Answer: ${?}+ 445 = 864$ ${445}$ ${864}$ $+?$ Let's start by adding hundreds to ${445}$ until we get as close to ${864}$ as possible without going over ${864}$. $\begin{aligned} {445} +100}=545\\\\ {545} +100}= 645\\\\ {645} +100}= 745\\\\ {745} +100}= 845 \end{aligned}$ If we add $4 \text{ hundreds}}$, or $4 00}$, we reach $845$. We cannot add any more hundreds without going over ${864}$. ${445}$ ${864}$ ${845}$ $+400$ Next, let's add tens to $845$ until we get as close to ${864}$ as possible without going over ${864}$. $\begin{aligned} 845 +{10}=855 \end{aligned}$ If we add ${1 \text{ ten}}$, or ${10}$, we reach $855$. We cannot add any more tens without going over ${864}$. ${445}$ ${864}$ ${845}$ ${855}$ $+400$ $+10$ Finally, how many ones should we add to $855$ to get to ${864}?$ $\begin{aligned} 855+{5} &=860\\\\ 860+{4} &=864 \end{aligned}$ We add ${9\text{ ones}}$. ${445}$ ${864}$ ${845}$ ${855}$ $+400$ $+10$ $+9$ We added $4 \text{ hundreds}}$, ${1 \text{ ten}}$, and ${9\text{ ones}}$ to ${445}$ to get to ${864}$. $4 00}+{1 0}+{9}={419}$ ${445}$ ${864}$ ${845}$ ${855}$ $+400$ $+10$ $+9$ $+419$ ${419}+ 445 = 864$